Gale Greenlee
Posted on Monday, 27 October, 2003 - 04:37 pm:
If I express Sq rt of 365/8 as a decimal it comes out approximately 6.754628. I think this is an irrational number by definition because that decimal equivalent never repeats and never terminates. Did I get that right?

I would like to divide my number by 2/3. As a decimal that is about 10.131891. How do I convert that to an expression like the sq rt of something?

GALE
Philip Ellison
Posted on Monday, 27 October, 2003 - 05:16 pm:

  _____
Ö365/8
 

is indeed irrational. The square root of any number that isn't a perfect square is irrational (this can be proved fairly easily; ask if you'd like me to show you how).


2/3=   ___
Ö4/9
 

. Therefore,
  _____
Ö365/8
 
 /(2/3) =   ___________
Ö(365/8)/(4/9)
 

Have you had any experience with manipulating surds?

James James
Posted on Monday, 27 October, 2003 - 05:46 pm:
Just to make the second part more clear, remember that
x = sqrt(x^2)
Therefore if x = 2/3, then
x^2 = 2/3 * 2/3 = 4/9
So 2/3 = sqrt(4/9)

For this question you also need to know that:
Sqrt(a) * Sqrt(b) = Sqrt(a*b)
and therefore also,
Sqrt(a)/Sqrt(b) = Sqrt(a/b)
Gale Greenlee
Posted on Monday, 27 October, 2003 - 07:45 pm:
I have had no experience manipulating surds.

GALE
James James
Posted on Monday, 27 October, 2003 - 08:25 pm:
Well, the what you need to remember is;
x = sqrt(x^2)
Sqrt(a) * Sqrt(b) = Sqrt(a*b)
Sqrt(a)/Sqrt(b) = Sqrt(a/b)

So, as a nice simple example,
sqrt(9) * sqrt(4) = 2 * 3 = sqrt(9*4) = sqrt(36) = 6

Also,
sqrt(8) = sqrt(2) * sqrt(4)
and since sqrt(4) = 2
sqrt(8) = 2 * sqrt(2)
So if you have a surd which can be split into factors which include square numbers, it can be simplified like this.

If you have a fraction containing surds in the denominator, for instance a/sqrt(b), then it often a good idea to rationalise the deniminator by multiplying by 1, in the form sqrt(b)/sqrt(b)
giving a*sqrt(b)/b. In the same way, if you have a fraction such as a/sqrt(b)+c, rationalise the denominator by multiplying by (sqrt(b)+c)/(sqrt(b)-c) to produce a difference of two squares as the denominator.

This is about all you need to know about surds.
It is also useful to know these now that you can manipulate surds;
sin 60 = sqrt(3)/2
Cos 30 = sqrt(3)/2
Sin 45 = 1/sqrt(2)
Cos 45 = 1/sqrt(2)
Sin 30 = 1/2
Cos 60 = 1/2

Hope this helps.
Gale Greenlee
Posted on Tuesday, 28 October, 2003 - 04:12 pm:
James:
Thanks so much. GALE